The Study Of Two-level Method And Additive Schwarz Method Based On AHSS Method

Two-level method is very efficient for solving nonlinear or nonselfadjoint and indefinite elliptic problems.By employing two subspaces,the solution of the original problem on the fine space is reduced to the solution of small problem on the coarse mesh and the solution of simpler problem on the fine mesh.So cost for solving the original problem can be reduced greatly.Therefore,this method has been widely applied to varieties of problems.For linear elliptic problems,domain decomposition methods can be viewed as preconditioners of the Krylov subspace method for the solution of linear systems arising from discretizing elliptic problems.Owing to the optimal convergence and good parallel properties,domain decomposition method has become one of the significant methods for solving the large-scale problems in engineer and science.Based on the work of predecessors,this paper will further study the applications of the two-level method and the Schwarz algorithm.This dissertation is divided into three chapters.In Chapter 1,the historical background and the current situation of the two-level method,the penalty method and the Schwarz algorithm are introduced.In addition,some function spaces and inequalities are also presented.In Chapter 2,we present a two-level penalty method for solving the steady state incompressible Stokes equations.By employing two subspaces,it is only needed to solve Stokes equations on the coarse space and two penalty parameters equations with the same coefficient matrix on the fine space.Note that the coefficient matrix is symmetric and positive.So the original Stokes equation can be solved easily.The convergence theory shows that the coarse space can be chosen very small corresponding to the fine space.And we can still obtain the optimal solution of the problem while the penalty parameter is not chosen very small.The numerical experiments also confirm the theory.Moreover,the numerical comparison also shows that the two-level penalty method has higher computational efficiency for solving the steady state incompressible Stokes equations.In Chapter 3,based on the AHSS(Asymmetric Hermitian and skew-Hermitian splitting)iteration method,a new two-level additive Schwarz algorithm applied to nonselfadjoint elliptic problems is proposed.We regard the Schwarz operator as a preconditioner of GMRES to solve the linear system deriving from the finite element discretization of the elliptic problem.And the parameters ? and ? in the Schwarz operator should be chosen as the maximum and minimum eigenvalues of Hermitian part of the coefficient matrix,respectively.The convergence theory of the new two-level additive Schwarz algorithm is presented by the classic Schwarz theory.Under certain assumptions,we prove that the convergence rate of this algorithm is bounded independent of the fine mash size and the number of subdomains.The numerical examples confirm our theory.Further,compared to the classic Schwarz algorithm,the new algorithm is more effective for convection-dominated diffusion equations.